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Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channel
Automatica 45 (2009) 2272–2277
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Brief paper
Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channelI Lei Zhou a , Guoping Lu b,∗ a
School of Science, Nantong University, Nantong, Jiangsu 226007, China
b
College of Electrical Engineering, Nantong University, Nantong, Jiangsu 226019, China
article
info
Article history: Received 21 October 2008 Received in revised form 22 February 2009 Accepted 26 May 2009 Available online 25 July 2009 Keywords: Detection Stabilization Discrete descriptor system Limited information Linear matrix inequality
abstract This paper addresses the state estimation and stabilization for linear discrete-time descriptor systems. It is assumed that the estimator and controller can only receive the transmitted sequence of finite coded signals via a limited digital communication channel. Both coder–decoder and coder–decoder–controller procedures are proposed, in which a compensation term is introduced to meet the compatible requirement. In addition, an effective method for selecting the compensation term is presented. Detection and stabilization conditions in terms of linear matrix inequality are obtained. Finally, an example is given to illustrate the effectiveness of design procedures. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction State estimation or state detection has received considerable attention in the last few decades. When not all the states of the controlled system are available for state feedback, a state observer is often required. Descriptor systems, also referred as singular systems, arise in many applications such as electrical circuits (Newcomb, 1981), constrained robot systems (McClamroch, 1986), etc. So far many results based on the theory of linear systems have been extended to the area of linear descriptor systems (Dai, 1989; Xu & James, 2006). Among these results, the problem of observer design for descriptor systems has been investigated in Darouach and Boutayeb (1995), Lu and Ho (2006) and Nikoukhah, Campbell, and Delebecque (1998). In classical estimation and control theory, a standard assumption is that the communication channel is perfect, that is, the measured output is the same as the input of the observer. However, this is not just the case in some practical situations, such as a large
I This work was supported by the National Natural Science Foundation of China under Grant No. 60874021, NSF Grant BK2007061 and ‘Liu Da Ren Cai’ Foundation of Jiangsu Province (No. 06-E-029). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Peterson. ∗ Corresponding author. Tel.: +86 513 85012609; fax: +86 513 85012600. E-mail addresses: [email protected] (L. Zhou), [email protected], [email protected] (G. Lu).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.05.022
number of mobile units need to be controlled remotely by a single decision maker (Savkin & Petersen, 2003) and controlling a platoon of autonomous underwater vehicles (Stilwell & Bishop, 2000). The situation may also arise in networked-based control systems, where the physical plant and observer/controller are located at different places and measured observation/control signals are sent via limited capacity communication networks. For state estimation and control problem via a limited capacity communication channel, there have been a few important results reported in the literature; see Schenato, Sinopoli, Franceschetti, Poolla, and Sastry (2007) for a brief review. Considering the bandwidth constraint of the communication channel, a modelbased networked control system architecture is introduced in Montestruque and Antsaklis (2003) to reduce the data transmission rate, further research is reported in Montestruque and Antsaklis (2007), where both static quantizer and dynamic quantizer are considered. Using uniform quantization method, a recursive coder–decoder state estimation scheme is proposed in Savkin and Petersen (2003), which is extended to the case of output feedback stabilization in Savkin (2007). H∞ estimation is reported in Gao and Chen (2007). It is worth mentioning that most of the previously mentioned researches were mainly concerned with non-descriptor systems; little attention has been paid to descriptor systems, though plentiful results have been established in classical state estimation and stabilization for descriptor systems (Lu & Jiang, 2006). The main difficulty may lie in the complexity of descriptor systems, such as compatible requirement of initial
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
conditions, the existence of the solution and impulsiveness of the state response. In this paper, we consider the state estimation and stabilization problem for linear discrete-time descriptor systems via a limited digital communication channel. It is assumed that observation and control signals are coded and sent via a communicationconstrained channel, the estimator/controller can only receive the transmitted sequence of finite coded signals. Since only the finitevalued signals are transmitted, classical theory cannot be applied to implement observation and stabilization problems (Matveev, 2008). By using uniform quantization method (Savkin, 2007) and introducing a compensation term, detectable and stabilizable conditions are obtained in this paper. The contributions of the paper are summarized as follows: (1) new coder–decoder and coder–decoder–controller procedures are designed to solve the state estimation and stabilization problem for the discrete-time descriptor systems via a limited digital communication channel, which are quite different from those of continuous-time system discussed in Lu and Jiang (2006) and Savkin (2007), see Remarks 3.2 and 3.3 for detail; (2) the derived detectable and stabilizable conditions are based on liner matrix inequalities, which are numerically efficient and simply verified. The remainder of the paper is organized as follows. The problem statements and some preliminaries are formulated in Section 2. The results for state estimation and stabilization are presented in Sections 3 and 4, respectively. An illustrative example is given in Section 5 and conclusion is presented in Section 6. Notation. Throughout this paper, Rn denotes the n-dimensional Euclidean space, Z+ is the set of positive integer. For x = (x1 x2 √ · · · xn )0 ∈ Rn and S ⊂ Rn , kxk = x0 x, kxk∞ = max{|xi |, 1 ≤ i ≤ n}, S − x = {y − x : y ∈ S }. Im (or I) is the m-dimensional (or appropriately dimensioned) identity matrix, W 0 denotes transpose of matrix W , W > 0 means that W is positive definite. Asterisk ∗ in a symmetric matrix denotes the entry implied by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem statements and preliminaries In this paper, we consider a linear discrete-time descriptor system of the form: Ex(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k), x(0) = x0 ∈ X0 ,
(1)
Fig. 1. State estimation via digital communication channel.
digital communication channel at each time jp. At the remote reception, an estimated state xˆ (k) or control input u(k) is produced by the received codeword h(jp). In particular, the detection procedure is illustrated in Fig. 1 and described as follows. jp
Coder : h(jp) = Fj (y(.) |0 ). (j+1)p
Decoder : xˆ (k) |jp
(2)
= Gj (h(p), h(2p), . . . h(jp))
(3)
for j = 1, 2, . . ., Fj and Gj are coder and decoder functions to be designed, respectively. For the stabilization problem, the decoder–controller procedure is as follows. Decoder–Controller: (j+1)p
u(k) |jp
= Gj (h(p), h(2p), . . . h(jp)).
(4)
Definition 2.1. System (1) is said to be detectable via a digital communication channel of capacity l if there exists a coder– decoder pair (2) and (3) with a coding alphabet of size l such that lim kx(k) − xˆ (k)k = 0
(5)
k→∞
for any solution of system (1). Definition 2.2. System (1) is said to be stabilizable via a digital communication channel of capacity l if there exists a coder– decoder–controller (2) and (4) with a coding alphabet of size l such that lim kx(k)k = 0,
k→∞
lim ku(k)k = 0
(6)
k→∞
for any solution of closed-loop system (1), (2) and (4). In the following, we will present two preliminary lemmas which are relevant in the designs of state estimation and stabilization procedure. Lemma 2.1. Given a scalar α > 0. Suppose that there exist positive definite matrix P > 0 and matrix R such that the following LMI is solvable 0 Ω := A0 PA + A0 E⊥ R0 + RE⊥ A − α 2 E 0 PE < 0,
where x ∈ R is the system state, u(k) ∈ R is the control input, y ∈ Rq is measured output. A, B and C are constant matrices of appropriate dimensions, x0 is a compatible initial condition and X0 is a known bounded set. Without loss of generality, suppose that 0 < rank(E ) = r < n. Throughout this paper, we use the pair (E , A) to denote the unforced descriptor system Ex(k + 1) = Ax(k) and make the following assumption. n
2273
m
Assumption 2.1. The pair (E , A) is regular. In our detection and stabilization problem, the state estimator/controller and the plant are separated by a communication channel with limited bandwidth. Since only a limited number of bits are available to the estimator, so a proper information encoding and decoding procedure is necessary. The procedure adopted in this paper is mainly inspired by Lu and Jiang (2006) and Savkin (2007). Generally speaking, for a known period p > 0, the encoded signal h(jp), which is selected from a coding alphabet H of size l and generated by the system output y(k), is transmitted through a
(7)
n×(n−r )
where E⊥ ∈ R is any matrix with full column rank and satisfies E 0 E⊥ = 0. Then there exists a constant µ > 0 such that
kx1 (k + 1) − x2 (k + 1)k∞ ≤ µαkx1 (k) − x2 (k)k∞
(8)
holds for any two solutions x1 (k) and x2 (k) of system (1). Proof. Let z (k) = α −k (x1 (k) − x2 (k)), then we have Ez (k + 1) = α −1 Az (k).
(9)
Choose the Lyapunov function V (k) = α z (k)E PEz (k), then, the difference of V (k) along system (9) is given by 2 0
0
∆V (k) = z (k)A0 PAz (k) − α 2 z 0 (k)E 0 PEz (k) + 2z 0 (k)A0 E⊥ R0 z (k)
= z 0 (k)Ω z (k) ≤ −λ0 kz (k)k2 ,
(10)
where λ0 = λmin (−Ω ) > 0. Denoting λm = λmax (α 2 E 0 PE ) > 0, it follows that V (k) ≤ λm kz (k)k2 and 1 −1 ∆V (k) ≤ −λ0 kz (k)k2 ≤ −λ0 λ− m V (k) ≤ −λ0 λm V (k + 1).
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Then for any x ∈ Ba , there exist unique integers i1 , i2 , . . . , in ∈ {1, 2, . . . , q} such that x ∈ Ii11 (a) × Ii22 (a) × · · · × Iinn (a), which implies that kx − ηa (i1 , i2 , . . . , in )k∞ ≤ qa . To present our coder–decoder procedure, we introduce the following set:
Therefore, V (k + 1) ≤ (1 + λ0 λm )
−1 −1
V (k) := β V (k),
where 0 < β = (1 + λ0 λm )
< 1. 1 Then denoting µ = (nλ0 λm β) 2 , we get −1 −1 −1
kz (k + 1)k∞ ≤ µkz (k)k∞ .
S (x) = {x ∈ Rn : 0
For z (k) = α −k (x1 (k) − x2 (k)), we have
where M is a nonsingular matrix satisfying MEN = diag{Ir , 0} together with nonsingular matrix N. Suppose that Lemmas 2.1 and 2.2 hold for some constants α and p, respectively. Furthermore, we define
kx1 (k + 1) − x2 (k + 1)k∞ ≤ µαkx1 (k) − x2 (k)k∞ . This completes the proof of Lemma 2.1.
Now consider the following Luenberger-like state observer that will be a part of proposed coder procedure: E x˜ (k + 1) = Ax˜ (k) − L(y(k) − C x˜ (k)) + Bu(k), x˜ (0) = x˜ 0 ∈ X0 ,
(11)
where x˜ (k) is the estimation of the system state x(k). The following lemma presents a sufficient condition for the design of state observer (11) by means of linear matrix inequality. Lemma 2.2. Suppose that there exist positive definite matrix Q > 0, matrices S, F and Y such that the following LMI is solvable:
Θ :=
Θ11 ∗
0 SE⊥ + A0 F + C 0 Y − F 0 Q − F − F0
In−r MAx = 0},
(15)
m0 = sup kx0 k∞ ; x0 ∈X0
a(p) = (2c + µp α p )m0 ; a((j + 1)p) = 2(c j+1 + 2µp α p c j )m0 + 2µp α p
δ(jp) = 2c j m0 +
a(jp) q
,
a(jp) q
;
(16)
j ≥ 1.
Now we present coder–decoder pairs as follows: Coder: For x˜ (jp) − x¯ (jp) ∈ Ii11 × Ii22 × · · · × Iinn ⊂ Ba(jp) , h(jp) = {i1 , i2 , . . . , in },
(17)
where x¯ (k) is defined by
< 0,
(12)
where Θ11 = A0 F + F 0 A + C 0 Y + Y 0 C − E 0 QE, E⊥ ∈ Rn×(n−r ) is given in Lemma 2.1, then there exist p ∈ Z+ and a constant c (0 < c < 1) such that
kx(k + p) − x˜ (k + p)k∞ ≤ c kx(k) − x˜ (k)k∞ .
(13)
Furthermore, the corresponding observer gain in (11) can be given as L = (YF −1 )0 . Proof. The proof is similar to that of Lemma 2.1 and is omitted here for space limitation.
x¯ (0) = 0; x¯ (k) = xˆ (k), k 6= jp; E x¯ (jp) = Axˆ (jp − 1).
(18)
Decoder: For h(jp) = {i1 , i2 , . . . , in }, xˆ (0) = 0; E xˆ (k + 1) = Axˆ (k), k 6= jp − 1; xˆ (jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp),
(19)
where
∆η(jp) ∈ Sjp := {∆η(jp) ∈ Bδ(jp) : x¯ (jp)
+ ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp) ∈ S (x)}.
(20)
3. State estimation via limited communication channel The purpose of this section is to develop coder–decoder procedure (2)–(3) and sufficient conditions for system (1) such that the corresponding estimator error satisfies condition (5). Without loss of generality, we assume that u(k) ≡ 0. In the following, we firstly recall the uniform state quantization method proposed by Savkin (2007). For a given scaling parameter a > 0 and an integer q, we can partition the hypercube Ba = {x ∈ Rn : kxk∞ ≤ a} into qn hypercubes Ii11 (a) × Ii22 (a) × · · · × Iinn (a), where i1 , i2 , . . . , in ∈ {1, 2, . . . , q} and I1i (a) := I2i (a) :=
.. . Iqi (a) :=
xi : −a ≤ xi < −a +
2a
xi : −a +
xi : a −
q
2a q
2a q
;
≤ x i < −a +
4a
q
; (14)
≤ xi ≤ a .
The center of the hypercube Ii11 (a) × Ii22 (a) × · · · × Iinn (a) is defined as
ηa (i1 , i2 , . . . , in ) =
−a +
(2i1 − 1)a
··· − a +
q
−a+
(2in − 1)a q
0
.
(2i2 − 1)a q
Remark 3.1. Different from non-descriptor system, the initial condition of descriptor system must be selected carefully such that the condition is compatible. In fact, S (x) is nothing but the set of compatible initial conditions, which is fundamental to the coder–decoder procedure for descriptor system. Remark 3.2. In the detection and stabilization problems for continuous-time system in Lu and Jiang (2006) and Savkin (2007), the limit state xˆ (jT − 0) is applied. However, one cannot define left limit xˆ (jp − 0) in discrete-time case, so auxiliary system (18) with state x¯ (k) is introduced in the above coder design procedure, which is the major difference from those of continuous-time system in Lu and Jiang (2006) and Savkin (2007). Remark 3.3. In decoder procedure (19), a compensation term ∆η(jp) is constructed which is a compatible requirement of descriptor systems for xˆ (jp) is used as the initial condition in the time period of jp ≤ k < (j + 1)p − 1. Generally speaking, ∆η(jp) 6= 0, the reason is that x¯ (jp) ∈ S (x) and ηa(jp) (i1 , i2 , . . . , in ) 6= 0 for the most cases. On the other hand, the term can also be regarded as a disturbance on the estimation state xˆ (jp). Therefore, proper choice of ∆η(jp) is fundamental to implement the coder–decoder procedure. The following lemma shows that the proposed coder–decoder procedure of (17)–(19) is well-posed. That is, the decoding condition x˜ (jp) − x¯ (jp) ∈ Ba(jp) holds and ∆η(jp) satisfying constraint (20) exists for all j = 1, 2, . . . .
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
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Lemma 3.1. For all j ≥ 1, coder–decoder procedure of (17)–(19) satisfies (i) x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) ∈ Sjp ; (ii) k˜x(jp) − x¯ (jp)k∞ ≤ a(jp).
Remark 3.5. If r = n − 1, then S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) is an (n − 1)-dimensional hyperplane. In this case, ∆η(jp) can be selected as the intersection point of the hyperplane and its normal line through the origin. So we have
Proof. The following inequality can be derived from inequality (13) and the decoder procedure:
∆η(jp) = −
kx(jp) − xˆ (jp)k∞ ≤ kx(jp) − x˜ (jp)k∞ + k˜x(jp) − xˆ (jp)k∞ ≤ c j kx(0) − x˜ (0)k∞ + k˜x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) − ∆η(jp)k∞ ≤ 2δ(jp).
q
(21)
= δ(jp).
(22)
Noticing x(jp) ∈ S (x), it follows from (22) that x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) ∈ Sjp for all j ≥ 1. In order to prove (ii), we apply the method of mathematical induction. For the case of j = 1, from (8) and (13), it follows that
k˜x(p) − x¯ (p)k∞ ≤ ≤ ≤ ≤
k˜x(p) − x(p)k∞ + kx(p) − x¯ (p)k∞ 2 cm0 + µαkx(p − 1) − xˆ (p − 1)k∞ 2 cm0 + µp α p kx(0) − xˆ (0)k∞ (2c + µp α p )m0 = a(p).
L0δ(jp) A0 M 0
0 0
0
(23)
In − r
MAη( ¯ jp)Lδ(jp)
,
Proof. Condition (26) and 0 < c < 1 imply that limj→∞ δ(jp) = 0. From (21), we conclude that lim kx(jp) − xˆ (jp)k∞ = 0.
j→∞
Given k ∈ Z+ , suppose that jp ≤ k ≤ (j + 1)p, we obtain
4. Stabilization via limited communication channel
Proposition 3.1. There exists a θ (jp) ∈ [−1, 1] such that ∆η(jp) defined below satisfies (20).
∆η(jp) =
holds for some positive integer q, where µ and p satisfy (8) and (13), respectively. Then system (1) is detectable via the coder–decoder procedure of (17)–(20).
lim kx(k) − xˆ (k)k = 0,
From the geometric viewpoint, the nonempty set Sjp is the intersection of the hypercube Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ). Therefore, there exists a point Lδ(jp) on one edge of the hypercube Bδ(jp) such that the line through Lδ(jp) and the origin intersects the set S (x)−¯x(jp)−ηa(jp) (i1 , i2 , . . . , in ) in the interior of Bδ(jp) , thus the intersection point can be selected as ∆η(jp) which can be calculated according to the following proposition.
In−r
(26)
Hence
Remark 3.4. The proof of Lemma 3.1 illustrates how the algorithm of the proposed coder–decoder procedure works and the set Sjp is not empty. One, however, cannot choose ∆η(jp) as x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) for x(jp) is an unknown system state to be estimated.
q > 2 µp α p
which completes the proof.
Therefore, (ii) holds for all j ≥ 1, which completes the proof.
0
Theorem 3.1. Given α > 0, suppose that linear matrix inequalities (7) and (12) are solvable. If the inequality
k→∞
≤ 2c j+1 m0 + 2µp α p δ(jp) = a((j + 1)p).
0 0
1 is an n-dimensional row vector.
i∈{0,...,p−1}
k˜x((j + 1)p) − x¯ ((j + 1)p)k∞ ≤ k˜x((j + 1)p) − x((j + 1)p)k∞ + kx((j + 1)p) − x¯ ((j + 1)p)k∞ ≤ 2c j+1 m0 + µp α p kx(jp) − xˆ (jp)k∞
0
(25)
kx(k) − xˆ (k)k∞ ≤ µk−jp α k−jp kx(jp) − xˆ (jp)k∞ ≤ max µi α i kx(jp) − xˆ (jp)k∞ .
Suppose that (ii) holds for 1, 2, . . . , j, then for j + 1, we have
−L0δ(jp) A0 M 0
···
,
Now we are in a position to present our main result of this section.
kx(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in )k∞ ≤ k˜x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in )k∞ + kx(jp) − x˜ (jp)k∞ ≤ 2c j m0 +
en MAA0 M 0 e0n
where en = 0
In addition,
a(jp)
en MAη¯ a(jp) A0 M 0 e0n
(24)
MALδ(jp)
where η( ¯ jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ), Lδ(jp) ∈ Rn has (n − 1) elements being either δ(jp) or −δ(jp) and one element being θ(jp)δ(jp).
This section is concerned with the stabilization problem for discrete-time descriptor system (1) via a limited capacity communication channel. It is assumed that E can be decomposed as E = EL ER 0 , where EL , ER ∈ Rn×r are of full column rank. Lemma 4.1 (Zhai, Koyama, Bruzelius, and Yoshida (2004)). Suppose that there exist nonsingular symmetric matrix X , matrices G and H, such that ER 0 XER > 0,
Ξ11 ∗
AG + BH − G0 X − G − G0
< 0,
(27)
where Ξ11 = AG + G0 A0 + BH + H 0 B0 − E 0 XE. Then the state feedback controller u(k) = HG−1 x(k) =: Kx(k) renders the closed-loop system of (1) globally asymptotically stable. We present the coder–decoder–controller pairs as follows: Coder: For x˜ (jp) − x¯ (jp) ∈ Ii11 × Ii22 × · · · × Iinn ⊂ Ba(jp) , h(jp) = {i1 , i2 , . . . , in },
(28)
where x¯ (k) is defined by x¯ (0) = 0; x¯ (k) = xˆ (k), k 6= jp; E x¯ (jp) = Axˆ (jp − 1) + Bu(jp − 1).
(29)
Decoder–Controller: For h(jp) = {i1 , i2 , . . . , in }, xˆ (0) = 0; E xˆ (k + 1) = Axˆ (k) + Bu(k), k 6= jp − 1; xˆ (jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp); u(k) = K xˆ (k),
(30)
x3
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L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1 –0.12 0.08
0.06
0.04
0.02
0
0.02
0 –0.02 –0.02 –0.04 –0.04 –0.06 –0.08 –0.08–0.06 x2 x
0.04
0.06
0.08
1
Fig. 2. Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , i3 ) for q = 60, j = 50. Fig. 3. The detection error state responses for q = 60.
where
∆η(jp) ∈ S˜jp := {∆η(jp) ∈ Bδ(jp) : x¯ (jp)
+ ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp) ∈ S˜(x)}, S˜(x) = {x ∈ R : 0 In−r M (A + BK )x = 0}
(31)
n
and a(jp), δ(jp) are defined in (16). Similarly to the state estimation problem in Section 3, we can get the following result for the feedback stabilization problem of system (1) via a communication channel with limited capacity. Theorem 4.1. Suppose that Lemma 4.1 holds, then under the conditions of Theorem 1, the coder–decoder–controller procedure of (28)–(31) asymptotically stabilizes system (1). Remark 4.1. Replacing A in (24) and (25) by A + BK , we can derive the corresponding formula to calculate ∆η(jp) in the decoder– controller procedure of (30) and (31). Fig. 4. The detection error state responses for q = 48.
5. A numerical example Example 5.1. Consider linear discrete-time descriptor system (1) with the following parameters E=
7 1 4
B= 3
6 18 0 1
2 −4 , 2
!
0
1 ,
A= C = 2
0 14 −1 3
13 7 4
12 −15 , 5
!
1 .
It is easy to show that rank(E ) = 2 and the pair (E , A) is regular. The generalized spectral radius of (E , A) is ρ(E , A) = max{|λ| : det(λE − A) = 0} = 1.0059 > 1, which implies that (E , A) is unstable (Xu & James, 2006). The compatible set defined in (15) is
S (x) = x = (x1 x2 x3 ) : −3x1 + 4x2 + 0
26 3
x3 = 0 .
(32)
Now we consider the state estimation problem, assume that u(k) ≡ 0. For α = 1.55, applying Lemma 2.1, we can get that (8) holds for µ = 1.8824. By using Lemma 2.2, the observer gain can be obtained as L = (−2.8433 −3.0245 −0.7772)0 . In addition, p = 3 and c = 0.8058 are obtained such that (13) holds. Therefore, we can get the detection condition by Theorem 3.1 is that q > 2µp α p = 49.6788. Choosing q = 60 > 49.6788 and m0 = 15. When j = 50, δ(jp) = 0.0661, using (25) to calculate ∆η(jp), we get that ∆η(jp) = (0.0150 −0.0201 −0.0434)0 . The relationship between Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , i3 ) is
shown in Fig. 2, where ‘∗’ denotes x(jp)− x¯ (jp)−ηa(jp) (i1 , i2 , , i3 ) = (0.0587 0.0552 −0.0631)0 . Applying the coder–decoder procedure of (17)–(19) with compatible set (32), the estimator errors between the state of the system and estimation state are shown in Fig. 3, where the initial conditions are x(0) = (3.6 11.67 −4.1400)0 and x˜ (0) = (9.6 1.4694 1.86)0 . Choosing q = 48 < 49.6788 and the same initial conditions, the estimator errors between the states of the system and estimation states are given in Fig. 4, which shows the error states are not convergent. Finally, we consider the stabilization problem and assume that u(k) 6= 0. By using Lemma 4.1 we get the state feedback gain K = (−1.5326 −4.7358 −1.8621). Applying the coder– decoder–controller procedure of (28)–(30) with q = 60 and m0 = 15, the state responses of the closed-loop system are shown in Fig. 5. 6. Conclusion In this paper, the problems of state estimation and stabilization via a limited communication channel for linear discretetime descriptor systems are considered. The coder–decoder and coder–decoder–controller procedures are proposed. It is worth mentioning that the explicit coder and decoder–controller design methods are established and the derived detectable and stabilizable conditions are numerically efficient and simply verified.
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
Fig. 5. The state responses of the closed-loop system for q = 60.
Furthermore, output feedback stabilization for nonlinear discretetime descriptor system is under consideration. References Dai, L. (1989). Singular control systems. Berlin, Germany: Springer-Verlag. Darouach, M., & Boutayeb, M. (1995). Design of observers for descriptor systems. IEEE Transactions on Automatic Control, 40(7), 1323–1327. Gao, H. J., & Chen, T. W. (2007). H∞ estimation for uncertain systems with limited communication capacity. IEEE Transactions on Automatic Control, 52(11), 2070–2084. Lu, G. P., & Ho, D. W. C. (2006). Full-order and reduced-order observers for Lipschitz descriptor systems: The unified LMI approach. IEEE Transactions on Circuits and Systems II- Express Briefs, 53(7), 563–567. Lu, G.P., & Jiang, Z.P. (2006). State estimation for networked descriptor systems with limited information. In Proceedings of the 25th Chinese control conference (pp. 1796–1801). Matveev, A. S. (2008). State estimation via limited capacity noisy communication channels. Mathematics of Control Signals and Systems, 20(1), 1–35. McClamroch, N. (1986). Singular systems of differential equations as dynamic models for constrained robot systems. In Proceedings of IEEE conference on robotics and automation (pp. 21–28).
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Montestruque, L. A., & Antsaklis, P. J. (2003). On the model-based control of networked systems. Automatica, 39(10), 1837–1843. Montestruque, L. A., & Antsaklis, P. J. (2007). Static and dynamic quantization in model-based networked control systems. International Journal of Control, 80(1), 87–101. Newcomb, R. W. (1981). The semistate description of nonlinear time variable circuits. IEEE Transactions on Circuits and Systems, 28(1), 62–71. Nikoukhah, R., Campbell, S. L., & Delebecque, F. (1998). Observer design for general linear time-invariant systems. Automatica, 34(5), 575–583. Savkin, A. V. (2007). Detectability and output feedback stabilizability of nonlinear networked control systems. IEEE Transactions on Automatic Control, 52(4), 730–735. Savkin, A. V., & Petersen, I. R. (2003). Set-valued state estimation via a limited capacity communication channel. IEEE Transactions on Automatic Control, 48(4), 676–680. Schenato, L., Sinopoli, B., Franceschetti, M., Poolla, K., & Sastry, S. S. (2007). Foundations of control and estimation over lossy networks. Proceedings of the IEEE, 95(1), 163–187. Stilwell, D. J., & Bishop, B. E. (2000). Platoons of underwater vehicles. IEEE Control Systems Magazine, 20(6), 45–52. Xu, S. Y., & James, Lam. (2006). Robust control and filtering of singular systems. Berlin, Germany: Springer-Verlag. Zhai, G.S., Koyama, N., Bruzelius, F., & Yoshida, M. (2004). Strict LMI conditions for stability and stabilization of discrete-time descriptor systems. In Proceedings of IEEE International symposium on intelligent control (pp. 495–499). Lei Zhou received the B.S. and M.S. degree from the Department of Mathematics, XuZhou Normal University, China, in 2001 and 2004, respectively. He joined Nantong University, Jiangsu, China, in 2004. Now he is a Ph.D. candidate of the Department of Mathematics, East China Normal University. His current research interests include nonlinear signal processing, descriptor system control and networked control.
Guoping Lu received the B.S. degree from the Department of Applied Mathematics, Chengdu University of Science and Technology, China, in 1984, and the M.S. and Ph.D. degrees from the Department of Mathematics, East China Normal University, China, in 1989 and 1998, respectively. He is currently a Professor at the College of Electrical Engineering, Nantong University, Jiangsu, China. His current research interests include nonlinear signal processing, robust control and networked control.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channelI Lei Zhou a , Guoping Lu b,∗ a
School of Science, Nantong University, Nantong, Jiangsu 226007, China
b
College of Electrical Engineering, Nantong University, Nantong, Jiangsu 226019, China
article
info
Article history: Received 21 October 2008 Received in revised form 22 February 2009 Accepted 26 May 2009 Available online 25 July 2009 Keywords: Detection Stabilization Discrete descriptor system Limited information Linear matrix inequality
abstract This paper addresses the state estimation and stabilization for linear discrete-time descriptor systems. It is assumed that the estimator and controller can only receive the transmitted sequence of finite coded signals via a limited digital communication channel. Both coder–decoder and coder–decoder–controller procedures are proposed, in which a compensation term is introduced to meet the compatible requirement. In addition, an effective method for selecting the compensation term is presented. Detection and stabilization conditions in terms of linear matrix inequality are obtained. Finally, an example is given to illustrate the effectiveness of design procedures. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction State estimation or state detection has received considerable attention in the last few decades. When not all the states of the controlled system are available for state feedback, a state observer is often required. Descriptor systems, also referred as singular systems, arise in many applications such as electrical circuits (Newcomb, 1981), constrained robot systems (McClamroch, 1986), etc. So far many results based on the theory of linear systems have been extended to the area of linear descriptor systems (Dai, 1989; Xu & James, 2006). Among these results, the problem of observer design for descriptor systems has been investigated in Darouach and Boutayeb (1995), Lu and Ho (2006) and Nikoukhah, Campbell, and Delebecque (1998). In classical estimation and control theory, a standard assumption is that the communication channel is perfect, that is, the measured output is the same as the input of the observer. However, this is not just the case in some practical situations, such as a large
I This work was supported by the National Natural Science Foundation of China under Grant No. 60874021, NSF Grant BK2007061 and ‘Liu Da Ren Cai’ Foundation of Jiangsu Province (No. 06-E-029). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Peterson. ∗ Corresponding author. Tel.: +86 513 85012609; fax: +86 513 85012600. E-mail addresses: [email protected] (L. Zhou), [email protected], [email protected] (G. Lu).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.05.022
number of mobile units need to be controlled remotely by a single decision maker (Savkin & Petersen, 2003) and controlling a platoon of autonomous underwater vehicles (Stilwell & Bishop, 2000). The situation may also arise in networked-based control systems, where the physical plant and observer/controller are located at different places and measured observation/control signals are sent via limited capacity communication networks. For state estimation and control problem via a limited capacity communication channel, there have been a few important results reported in the literature; see Schenato, Sinopoli, Franceschetti, Poolla, and Sastry (2007) for a brief review. Considering the bandwidth constraint of the communication channel, a modelbased networked control system architecture is introduced in Montestruque and Antsaklis (2003) to reduce the data transmission rate, further research is reported in Montestruque and Antsaklis (2007), where both static quantizer and dynamic quantizer are considered. Using uniform quantization method, a recursive coder–decoder state estimation scheme is proposed in Savkin and Petersen (2003), which is extended to the case of output feedback stabilization in Savkin (2007). H∞ estimation is reported in Gao and Chen (2007). It is worth mentioning that most of the previously mentioned researches were mainly concerned with non-descriptor systems; little attention has been paid to descriptor systems, though plentiful results have been established in classical state estimation and stabilization for descriptor systems (Lu & Jiang, 2006). The main difficulty may lie in the complexity of descriptor systems, such as compatible requirement of initial
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
conditions, the existence of the solution and impulsiveness of the state response. In this paper, we consider the state estimation and stabilization problem for linear discrete-time descriptor systems via a limited digital communication channel. It is assumed that observation and control signals are coded and sent via a communicationconstrained channel, the estimator/controller can only receive the transmitted sequence of finite coded signals. Since only the finitevalued signals are transmitted, classical theory cannot be applied to implement observation and stabilization problems (Matveev, 2008). By using uniform quantization method (Savkin, 2007) and introducing a compensation term, detectable and stabilizable conditions are obtained in this paper. The contributions of the paper are summarized as follows: (1) new coder–decoder and coder–decoder–controller procedures are designed to solve the state estimation and stabilization problem for the discrete-time descriptor systems via a limited digital communication channel, which are quite different from those of continuous-time system discussed in Lu and Jiang (2006) and Savkin (2007), see Remarks 3.2 and 3.3 for detail; (2) the derived detectable and stabilizable conditions are based on liner matrix inequalities, which are numerically efficient and simply verified. The remainder of the paper is organized as follows. The problem statements and some preliminaries are formulated in Section 2. The results for state estimation and stabilization are presented in Sections 3 and 4, respectively. An illustrative example is given in Section 5 and conclusion is presented in Section 6. Notation. Throughout this paper, Rn denotes the n-dimensional Euclidean space, Z+ is the set of positive integer. For x = (x1 x2 √ · · · xn )0 ∈ Rn and S ⊂ Rn , kxk = x0 x, kxk∞ = max{|xi |, 1 ≤ i ≤ n}, S − x = {y − x : y ∈ S }. Im (or I) is the m-dimensional (or appropriately dimensioned) identity matrix, W 0 denotes transpose of matrix W , W > 0 means that W is positive definite. Asterisk ∗ in a symmetric matrix denotes the entry implied by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem statements and preliminaries In this paper, we consider a linear discrete-time descriptor system of the form: Ex(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k), x(0) = x0 ∈ X0 ,
(1)
Fig. 1. State estimation via digital communication channel.
digital communication channel at each time jp. At the remote reception, an estimated state xˆ (k) or control input u(k) is produced by the received codeword h(jp). In particular, the detection procedure is illustrated in Fig. 1 and described as follows. jp
Coder : h(jp) = Fj (y(.) |0 ). (j+1)p
Decoder : xˆ (k) |jp
(2)
= Gj (h(p), h(2p), . . . h(jp))
(3)
for j = 1, 2, . . ., Fj and Gj are coder and decoder functions to be designed, respectively. For the stabilization problem, the decoder–controller procedure is as follows. Decoder–Controller: (j+1)p
u(k) |jp
= Gj (h(p), h(2p), . . . h(jp)).
(4)
Definition 2.1. System (1) is said to be detectable via a digital communication channel of capacity l if there exists a coder– decoder pair (2) and (3) with a coding alphabet of size l such that lim kx(k) − xˆ (k)k = 0
(5)
k→∞
for any solution of system (1). Definition 2.2. System (1) is said to be stabilizable via a digital communication channel of capacity l if there exists a coder– decoder–controller (2) and (4) with a coding alphabet of size l such that lim kx(k)k = 0,
k→∞
lim ku(k)k = 0
(6)
k→∞
for any solution of closed-loop system (1), (2) and (4). In the following, we will present two preliminary lemmas which are relevant in the designs of state estimation and stabilization procedure. Lemma 2.1. Given a scalar α > 0. Suppose that there exist positive definite matrix P > 0 and matrix R such that the following LMI is solvable 0 Ω := A0 PA + A0 E⊥ R0 + RE⊥ A − α 2 E 0 PE < 0,
where x ∈ R is the system state, u(k) ∈ R is the control input, y ∈ Rq is measured output. A, B and C are constant matrices of appropriate dimensions, x0 is a compatible initial condition and X0 is a known bounded set. Without loss of generality, suppose that 0 < rank(E ) = r < n. Throughout this paper, we use the pair (E , A) to denote the unforced descriptor system Ex(k + 1) = Ax(k) and make the following assumption. n
2273
m
Assumption 2.1. The pair (E , A) is regular. In our detection and stabilization problem, the state estimator/controller and the plant are separated by a communication channel with limited bandwidth. Since only a limited number of bits are available to the estimator, so a proper information encoding and decoding procedure is necessary. The procedure adopted in this paper is mainly inspired by Lu and Jiang (2006) and Savkin (2007). Generally speaking, for a known period p > 0, the encoded signal h(jp), which is selected from a coding alphabet H of size l and generated by the system output y(k), is transmitted through a
(7)
n×(n−r )
where E⊥ ∈ R is any matrix with full column rank and satisfies E 0 E⊥ = 0. Then there exists a constant µ > 0 such that
kx1 (k + 1) − x2 (k + 1)k∞ ≤ µαkx1 (k) − x2 (k)k∞
(8)
holds for any two solutions x1 (k) and x2 (k) of system (1). Proof. Let z (k) = α −k (x1 (k) − x2 (k)), then we have Ez (k + 1) = α −1 Az (k).
(9)
Choose the Lyapunov function V (k) = α z (k)E PEz (k), then, the difference of V (k) along system (9) is given by 2 0
0
∆V (k) = z (k)A0 PAz (k) − α 2 z 0 (k)E 0 PEz (k) + 2z 0 (k)A0 E⊥ R0 z (k)
= z 0 (k)Ω z (k) ≤ −λ0 kz (k)k2 ,
(10)
where λ0 = λmin (−Ω ) > 0. Denoting λm = λmax (α 2 E 0 PE ) > 0, it follows that V (k) ≤ λm kz (k)k2 and 1 −1 ∆V (k) ≤ −λ0 kz (k)k2 ≤ −λ0 λ− m V (k) ≤ −λ0 λm V (k + 1).
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L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
Then for any x ∈ Ba , there exist unique integers i1 , i2 , . . . , in ∈ {1, 2, . . . , q} such that x ∈ Ii11 (a) × Ii22 (a) × · · · × Iinn (a), which implies that kx − ηa (i1 , i2 , . . . , in )k∞ ≤ qa . To present our coder–decoder procedure, we introduce the following set:
Therefore, V (k + 1) ≤ (1 + λ0 λm )
−1 −1
V (k) := β V (k),
where 0 < β = (1 + λ0 λm )
< 1. 1 Then denoting µ = (nλ0 λm β) 2 , we get −1 −1 −1
kz (k + 1)k∞ ≤ µkz (k)k∞ .
S (x) = {x ∈ Rn : 0
For z (k) = α −k (x1 (k) − x2 (k)), we have
where M is a nonsingular matrix satisfying MEN = diag{Ir , 0} together with nonsingular matrix N. Suppose that Lemmas 2.1 and 2.2 hold for some constants α and p, respectively. Furthermore, we define
kx1 (k + 1) − x2 (k + 1)k∞ ≤ µαkx1 (k) − x2 (k)k∞ . This completes the proof of Lemma 2.1.
Now consider the following Luenberger-like state observer that will be a part of proposed coder procedure: E x˜ (k + 1) = Ax˜ (k) − L(y(k) − C x˜ (k)) + Bu(k), x˜ (0) = x˜ 0 ∈ X0 ,
(11)
where x˜ (k) is the estimation of the system state x(k). The following lemma presents a sufficient condition for the design of state observer (11) by means of linear matrix inequality. Lemma 2.2. Suppose that there exist positive definite matrix Q > 0, matrices S, F and Y such that the following LMI is solvable:
Θ :=
Θ11 ∗
0 SE⊥ + A0 F + C 0 Y − F 0 Q − F − F0
In−r MAx = 0},
(15)
m0 = sup kx0 k∞ ; x0 ∈X0
a(p) = (2c + µp α p )m0 ; a((j + 1)p) = 2(c j+1 + 2µp α p c j )m0 + 2µp α p
δ(jp) = 2c j m0 +
a(jp) q
,
a(jp) q
;
(16)
j ≥ 1.
Now we present coder–decoder pairs as follows: Coder: For x˜ (jp) − x¯ (jp) ∈ Ii11 × Ii22 × · · · × Iinn ⊂ Ba(jp) , h(jp) = {i1 , i2 , . . . , in },
(17)
where x¯ (k) is defined by
< 0,
(12)
where Θ11 = A0 F + F 0 A + C 0 Y + Y 0 C − E 0 QE, E⊥ ∈ Rn×(n−r ) is given in Lemma 2.1, then there exist p ∈ Z+ and a constant c (0 < c < 1) such that
kx(k + p) − x˜ (k + p)k∞ ≤ c kx(k) − x˜ (k)k∞ .
(13)
Furthermore, the corresponding observer gain in (11) can be given as L = (YF −1 )0 . Proof. The proof is similar to that of Lemma 2.1 and is omitted here for space limitation.
x¯ (0) = 0; x¯ (k) = xˆ (k), k 6= jp; E x¯ (jp) = Axˆ (jp − 1).
(18)
Decoder: For h(jp) = {i1 , i2 , . . . , in }, xˆ (0) = 0; E xˆ (k + 1) = Axˆ (k), k 6= jp − 1; xˆ (jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp),
(19)
where
∆η(jp) ∈ Sjp := {∆η(jp) ∈ Bδ(jp) : x¯ (jp)
+ ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp) ∈ S (x)}.
(20)
3. State estimation via limited communication channel The purpose of this section is to develop coder–decoder procedure (2)–(3) and sufficient conditions for system (1) such that the corresponding estimator error satisfies condition (5). Without loss of generality, we assume that u(k) ≡ 0. In the following, we firstly recall the uniform state quantization method proposed by Savkin (2007). For a given scaling parameter a > 0 and an integer q, we can partition the hypercube Ba = {x ∈ Rn : kxk∞ ≤ a} into qn hypercubes Ii11 (a) × Ii22 (a) × · · · × Iinn (a), where i1 , i2 , . . . , in ∈ {1, 2, . . . , q} and I1i (a) := I2i (a) :=
.. . Iqi (a) :=
xi : −a ≤ xi < −a +
2a
xi : −a +
xi : a −
q
2a q
2a q
;
≤ x i < −a +
4a
q
; (14)
≤ xi ≤ a .
The center of the hypercube Ii11 (a) × Ii22 (a) × · · · × Iinn (a) is defined as
ηa (i1 , i2 , . . . , in ) =
−a +
(2i1 − 1)a
··· − a +
q
−a+
(2in − 1)a q
0
.
(2i2 − 1)a q
Remark 3.1. Different from non-descriptor system, the initial condition of descriptor system must be selected carefully such that the condition is compatible. In fact, S (x) is nothing but the set of compatible initial conditions, which is fundamental to the coder–decoder procedure for descriptor system. Remark 3.2. In the detection and stabilization problems for continuous-time system in Lu and Jiang (2006) and Savkin (2007), the limit state xˆ (jT − 0) is applied. However, one cannot define left limit xˆ (jp − 0) in discrete-time case, so auxiliary system (18) with state x¯ (k) is introduced in the above coder design procedure, which is the major difference from those of continuous-time system in Lu and Jiang (2006) and Savkin (2007). Remark 3.3. In decoder procedure (19), a compensation term ∆η(jp) is constructed which is a compatible requirement of descriptor systems for xˆ (jp) is used as the initial condition in the time period of jp ≤ k < (j + 1)p − 1. Generally speaking, ∆η(jp) 6= 0, the reason is that x¯ (jp) ∈ S (x) and ηa(jp) (i1 , i2 , . . . , in ) 6= 0 for the most cases. On the other hand, the term can also be regarded as a disturbance on the estimation state xˆ (jp). Therefore, proper choice of ∆η(jp) is fundamental to implement the coder–decoder procedure. The following lemma shows that the proposed coder–decoder procedure of (17)–(19) is well-posed. That is, the decoding condition x˜ (jp) − x¯ (jp) ∈ Ba(jp) holds and ∆η(jp) satisfying constraint (20) exists for all j = 1, 2, . . . .
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
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Lemma 3.1. For all j ≥ 1, coder–decoder procedure of (17)–(19) satisfies (i) x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) ∈ Sjp ; (ii) k˜x(jp) − x¯ (jp)k∞ ≤ a(jp).
Remark 3.5. If r = n − 1, then S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) is an (n − 1)-dimensional hyperplane. In this case, ∆η(jp) can be selected as the intersection point of the hyperplane and its normal line through the origin. So we have
Proof. The following inequality can be derived from inequality (13) and the decoder procedure:
∆η(jp) = −
kx(jp) − xˆ (jp)k∞ ≤ kx(jp) − x˜ (jp)k∞ + k˜x(jp) − xˆ (jp)k∞ ≤ c j kx(0) − x˜ (0)k∞ + k˜x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) − ∆η(jp)k∞ ≤ 2δ(jp).
q
(21)
= δ(jp).
(22)
Noticing x(jp) ∈ S (x), it follows from (22) that x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) ∈ Sjp for all j ≥ 1. In order to prove (ii), we apply the method of mathematical induction. For the case of j = 1, from (8) and (13), it follows that
k˜x(p) − x¯ (p)k∞ ≤ ≤ ≤ ≤
k˜x(p) − x(p)k∞ + kx(p) − x¯ (p)k∞ 2 cm0 + µαkx(p − 1) − xˆ (p − 1)k∞ 2 cm0 + µp α p kx(0) − xˆ (0)k∞ (2c + µp α p )m0 = a(p).
L0δ(jp) A0 M 0
0 0
0
(23)
In − r
MAη( ¯ jp)Lδ(jp)
,
Proof. Condition (26) and 0 < c < 1 imply that limj→∞ δ(jp) = 0. From (21), we conclude that lim kx(jp) − xˆ (jp)k∞ = 0.
j→∞
Given k ∈ Z+ , suppose that jp ≤ k ≤ (j + 1)p, we obtain
4. Stabilization via limited communication channel
Proposition 3.1. There exists a θ (jp) ∈ [−1, 1] such that ∆η(jp) defined below satisfies (20).
∆η(jp) =
holds for some positive integer q, where µ and p satisfy (8) and (13), respectively. Then system (1) is detectable via the coder–decoder procedure of (17)–(20).
lim kx(k) − xˆ (k)k = 0,
From the geometric viewpoint, the nonempty set Sjp is the intersection of the hypercube Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ). Therefore, there exists a point Lδ(jp) on one edge of the hypercube Bδ(jp) such that the line through Lδ(jp) and the origin intersects the set S (x)−¯x(jp)−ηa(jp) (i1 , i2 , . . . , in ) in the interior of Bδ(jp) , thus the intersection point can be selected as ∆η(jp) which can be calculated according to the following proposition.
In−r
(26)
Hence
Remark 3.4. The proof of Lemma 3.1 illustrates how the algorithm of the proposed coder–decoder procedure works and the set Sjp is not empty. One, however, cannot choose ∆η(jp) as x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in ) for x(jp) is an unknown system state to be estimated.
q > 2 µp α p
which completes the proof.
Therefore, (ii) holds for all j ≥ 1, which completes the proof.
0
Theorem 3.1. Given α > 0, suppose that linear matrix inequalities (7) and (12) are solvable. If the inequality
k→∞
≤ 2c j+1 m0 + 2µp α p δ(jp) = a((j + 1)p).
0 0
1 is an n-dimensional row vector.
i∈{0,...,p−1}
k˜x((j + 1)p) − x¯ ((j + 1)p)k∞ ≤ k˜x((j + 1)p) − x((j + 1)p)k∞ + kx((j + 1)p) − x¯ ((j + 1)p)k∞ ≤ 2c j+1 m0 + µp α p kx(jp) − xˆ (jp)k∞
0
(25)
kx(k) − xˆ (k)k∞ ≤ µk−jp α k−jp kx(jp) − xˆ (jp)k∞ ≤ max µi α i kx(jp) − xˆ (jp)k∞ .
Suppose that (ii) holds for 1, 2, . . . , j, then for j + 1, we have
−L0δ(jp) A0 M 0
···
,
Now we are in a position to present our main result of this section.
kx(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in )k∞ ≤ k˜x(jp) − x¯ (jp) − ηa(jp) (i1 , i2 , . . . , in )k∞ + kx(jp) − x˜ (jp)k∞ ≤ 2c j m0 +
en MAA0 M 0 e0n
where en = 0
In addition,
a(jp)
en MAη¯ a(jp) A0 M 0 e0n
(24)
MALδ(jp)
where η( ¯ jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ), Lδ(jp) ∈ Rn has (n − 1) elements being either δ(jp) or −δ(jp) and one element being θ(jp)δ(jp).
This section is concerned with the stabilization problem for discrete-time descriptor system (1) via a limited capacity communication channel. It is assumed that E can be decomposed as E = EL ER 0 , where EL , ER ∈ Rn×r are of full column rank. Lemma 4.1 (Zhai, Koyama, Bruzelius, and Yoshida (2004)). Suppose that there exist nonsingular symmetric matrix X , matrices G and H, such that ER 0 XER > 0,
Ξ11 ∗
AG + BH − G0 X − G − G0
< 0,
(27)
where Ξ11 = AG + G0 A0 + BH + H 0 B0 − E 0 XE. Then the state feedback controller u(k) = HG−1 x(k) =: Kx(k) renders the closed-loop system of (1) globally asymptotically stable. We present the coder–decoder–controller pairs as follows: Coder: For x˜ (jp) − x¯ (jp) ∈ Ii11 × Ii22 × · · · × Iinn ⊂ Ba(jp) , h(jp) = {i1 , i2 , . . . , in },
(28)
where x¯ (k) is defined by x¯ (0) = 0; x¯ (k) = xˆ (k), k 6= jp; E x¯ (jp) = Axˆ (jp − 1) + Bu(jp − 1).
(29)
Decoder–Controller: For h(jp) = {i1 , i2 , . . . , in }, xˆ (0) = 0; E xˆ (k + 1) = Axˆ (k) + Bu(k), k 6= jp − 1; xˆ (jp) = x¯ (jp) + ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp); u(k) = K xˆ (k),
(30)
x3
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L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1 –0.12 0.08
0.06
0.04
0.02
0
0.02
0 –0.02 –0.02 –0.04 –0.04 –0.06 –0.08 –0.08–0.06 x2 x
0.04
0.06
0.08
1
Fig. 2. Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , i3 ) for q = 60, j = 50. Fig. 3. The detection error state responses for q = 60.
where
∆η(jp) ∈ S˜jp := {∆η(jp) ∈ Bδ(jp) : x¯ (jp)
+ ηa(jp) (i1 , i2 , . . . , in ) + ∆η(jp) ∈ S˜(x)}, S˜(x) = {x ∈ R : 0 In−r M (A + BK )x = 0}
(31)
n
and a(jp), δ(jp) are defined in (16). Similarly to the state estimation problem in Section 3, we can get the following result for the feedback stabilization problem of system (1) via a communication channel with limited capacity. Theorem 4.1. Suppose that Lemma 4.1 holds, then under the conditions of Theorem 1, the coder–decoder–controller procedure of (28)–(31) asymptotically stabilizes system (1). Remark 4.1. Replacing A in (24) and (25) by A + BK , we can derive the corresponding formula to calculate ∆η(jp) in the decoder– controller procedure of (30) and (31). Fig. 4. The detection error state responses for q = 48.
5. A numerical example Example 5.1. Consider linear discrete-time descriptor system (1) with the following parameters E=
7 1 4
B= 3
6 18 0 1
2 −4 , 2
!
0
1 ,
A= C = 2
0 14 −1 3
13 7 4
12 −15 , 5
!
1 .
It is easy to show that rank(E ) = 2 and the pair (E , A) is regular. The generalized spectral radius of (E , A) is ρ(E , A) = max{|λ| : det(λE − A) = 0} = 1.0059 > 1, which implies that (E , A) is unstable (Xu & James, 2006). The compatible set defined in (15) is
S (x) = x = (x1 x2 x3 ) : −3x1 + 4x2 + 0
26 3
x3 = 0 .
(32)
Now we consider the state estimation problem, assume that u(k) ≡ 0. For α = 1.55, applying Lemma 2.1, we can get that (8) holds for µ = 1.8824. By using Lemma 2.2, the observer gain can be obtained as L = (−2.8433 −3.0245 −0.7772)0 . In addition, p = 3 and c = 0.8058 are obtained such that (13) holds. Therefore, we can get the detection condition by Theorem 3.1 is that q > 2µp α p = 49.6788. Choosing q = 60 > 49.6788 and m0 = 15. When j = 50, δ(jp) = 0.0661, using (25) to calculate ∆η(jp), we get that ∆η(jp) = (0.0150 −0.0201 −0.0434)0 . The relationship between Bδ(jp) and S (x) − x¯ (jp) − ηa(jp) (i1 , i2 , i3 ) is
shown in Fig. 2, where ‘∗’ denotes x(jp)− x¯ (jp)−ηa(jp) (i1 , i2 , , i3 ) = (0.0587 0.0552 −0.0631)0 . Applying the coder–decoder procedure of (17)–(19) with compatible set (32), the estimator errors between the state of the system and estimation state are shown in Fig. 3, where the initial conditions are x(0) = (3.6 11.67 −4.1400)0 and x˜ (0) = (9.6 1.4694 1.86)0 . Choosing q = 48 < 49.6788 and the same initial conditions, the estimator errors between the states of the system and estimation states are given in Fig. 4, which shows the error states are not convergent. Finally, we consider the stabilization problem and assume that u(k) 6= 0. By using Lemma 4.1 we get the state feedback gain K = (−1.5326 −4.7358 −1.8621). Applying the coder– decoder–controller procedure of (28)–(30) with q = 60 and m0 = 15, the state responses of the closed-loop system are shown in Fig. 5. 6. Conclusion In this paper, the problems of state estimation and stabilization via a limited communication channel for linear discretetime descriptor systems are considered. The coder–decoder and coder–decoder–controller procedures are proposed. It is worth mentioning that the explicit coder and decoder–controller design methods are established and the derived detectable and stabilizable conditions are numerically efficient and simply verified.
L. Zhou, G. Lu / Automatica 45 (2009) 2272–2277
Fig. 5. The state responses of the closed-loop system for q = 60.
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Guoping Lu received the B.S. degree from the Department of Applied Mathematics, Chengdu University of Science and Technology, China, in 1984, and the M.S. and Ph.D. degrees from the Department of Mathematics, East China Normal University, China, in 1989 and 1998, respectively. He is currently a Professor at the College of Electrical Engineering, Nantong University, Jiangsu, China. His current research interests include nonlinear signal processing, robust control and networked control.